To identify the function [tex]\( h(x) \)[/tex] as a transformation of the square root parent function [tex]\( f(x) = \sqrt{x} \)[/tex], we need to consider the possible transformations that can be applied to [tex]\( f(x) \)[/tex]. These transformations can include vertical shifts, horizontal shifts, and other more complex operations.
Let's analyze each given option based on transformations and match it with a logical step-by-step understanding.
A. [tex]\( h(x) = \sqrt{x} - 5 \)[/tex]:
- This represents a vertical shift downward by 5 units. If [tex]\( f(x) = \sqrt{x} \)[/tex], then [tex]\( h(x) = f(x) - 5 \)[/tex] would translate the graph 5 units down.
B. [tex]\( h(x) = \sqrt{x+5} \)[/tex]:
- This represents a horizontal shift to the left by 5 units. If [tex]\( f(x) = \sqrt{x} \)[/tex], then [tex]\( h(x) = f(x + 5) \)[/tex] would translate the graph 5 units to the left.
C. [tex]\( h(x) = \sqrt{x} + 5 \)[/tex]:
- This represents a vertical shift upward by 5 units. If [tex]\( f(x) = \sqrt{x} \)[/tex], then [tex]\( h(x) = f(x) + 5 \)[/tex] would translate the graph 5 units up.
D. [tex]\( h(x) = \sqrt{x - 5} \)[/tex]:
- This represents a horizontal shift to the right by 5 units. If [tex]\( f(x) = \sqrt{x} \)[/tex], then [tex]\( h(x) = f(x - 5) \)[/tex] would translate the graph 5 units to the right.
We are tasked with identifying the transformation that matches the function [tex]\( h(x) \)[/tex]. By understanding the fundamental transformations:
The correct transformation of the function [tex]\( f(x) = \sqrt{x} \)[/tex] specifying the transformation of [tex]\( x \)[/tex] by subtracting 5 falls under the horizontal translation to the right by 5 units, which corresponds to:
[tex]\[ h(x) = \sqrt{x - 5} \][/tex]
Thus, the correct function is:
[tex]\[ \boxed{h(x) = \sqrt{x - 5}} \][/tex]
